S-AI-CCC: Controlled Cognitive Canonicalization

Abstract

Background. Large language models routinely transform mathematical expressions, logical formulas, symbolic problems, and unstructured inputs into more regular, compressed, or conventionally normalized forms. We call this phenomenon implicit canonicalization: an emergent statistical tendency, effective for discovering structurally meaningful candidates but carrying no guarantee of equivalence, canonical membership, or logical validity. The gap addressed here is the structural transition from probabilistic discovery to formally controlled decision. Methods. We introduce S-AI-CCC, a controlled cognitive canonicalization framework within Sparse Artificial Intelligence (S-AI). It separates and coordinates three functions along the pipeline x → LLM → y → S-AI-Recursive → y′ → S-AI-RLM → (y, v): candidate generation by an LLM, recursive stabilization by S-AI-Recursive, and terminal verification by S-AI-RLM, where v* ∈ {Accept, Reject, Clarify/Abstain} and a canonical output is committed only on the Accept branch. S-AI-Recursive models canonicalization as a closed-loop cognitive dynamics regulated by two antagonistic hormones, Clarifine and Confusionin; convergence is established under explicit dissipation, cross-inhibition, task-switch, emission, and delay conditions via a delay-dependent Halanay–Lyapunov estimate. A semantic parser certifies entry into a computable basin set contained in the correct input-dependent attractor, and S-AI-RLM applies a total verifier within a finite verification budget on the admissible bounded domain.Results. The framework establishes six regime-specific, assumption-explicit results. Deterministic exponential stability holds under the full deployability and delay conditions, while persistent reflected diffusion yields mean-square ultimate boundedness. A conditional coupling relates Lyapunov contraction to entropy contraction, and a Gibbs-family relation links decreasing entropy to increasing expected coherence for a fixed candidate family. Finite deterministic termination follows from an oriented hormonal stopping rule together with the verifier budget, and a joint Decidability–Convergence guarantee shows that hormonal convergence and a certified terminal verdict follow from a single explicit assumption set. Autonomous post-initialization dynamics are exponentially stable, whereas continuously LLM-driven dynamics are input-to-state stable, with bounded persistent inputs producing a corresponding ultimate bound. Conclusions. S-AI-CCC defines a class of Controlled Canonical Reasoning Systems in which the LLM discovers candidates, the regulated recursive layer stabilizes the cognitive trajectory, and the logical machine decides whether a candidate may be committed. Convergence, coherence, and decidability are coordinated but remain mathematically distinct. The framework thereby converts implicit canonicalization from an uncontrolled statistical tendency into a controlled cognitive process that accepts, rejects, clarifies, or abstains under explicit deterministic, stochastic, basin, and verifier assumptions, preserving the usefulness of generative discovery while relocating guarantees to the layers able to support them: stability to the regulated dynamics, probabilistic ordering to the stated distributional model, and correctness to the total verifier.